from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(87362, base_ring=CyclotomicField(3762))
M = H._module
chi = DirichletCharacter(H, M([3420,2959]))
pari: [g,chi] = znchar(Mod(67,87362))
Basic properties
| Modulus: | \(87362\) | |
| Conductor: | \(43681\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(3762\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{43681}(67,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 87362.di
\(\chi_{87362}(67,\cdot)\) \(\chi_{87362}(89,\cdot)\) \(\chi_{87362}(155,\cdot)\) \(\chi_{87362}(287,\cdot)\) \(\chi_{87362}(375,\cdot)\) \(\chi_{87362}(507,\cdot)\) \(\chi_{87362}(573,\cdot)\) \(\chi_{87362}(661,\cdot)\) \(\chi_{87362}(705,\cdot)\) \(\chi_{87362}(793,\cdot)\) \(\chi_{87362}(903,\cdot)\) \(\chi_{87362}(925,\cdot)\) \(\chi_{87362}(991,\cdot)\) \(\chi_{87362}(1079,\cdot)\) \(\chi_{87362}(1123,\cdot)\) \(\chi_{87362}(1321,\cdot)\) \(\chi_{87362}(1343,\cdot)\) \(\chi_{87362}(1409,\cdot)\) \(\chi_{87362}(1497,\cdot)\) \(\chi_{87362}(1541,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | $\Q(\zeta_{1881})$ |
| Fixed field: | Number field defined by a degree 3762 polynomial (not computed) |
Values on generators
\((21661,22023)\) → \((e\left(\frac{10}{11}\right),e\left(\frac{269}{342}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 87362 }(67, a) \) | \(-1\) | \(1\) | \(e\left(\frac{113}{342}\right)\) | \(e\left(\frac{1415}{1881}\right)\) | \(e\left(\frac{217}{627}\right)\) | \(e\left(\frac{113}{171}\right)\) | \(e\left(\frac{2231}{3762}\right)\) | \(e\left(\frac{311}{3762}\right)\) | \(e\left(\frac{575}{1881}\right)\) | \(e\left(\frac{2545}{3762}\right)\) | \(e\left(\frac{889}{1881}\right)\) | \(e\left(\frac{949}{1881}\right)\) |
sage: chi.jacobi_sum(n)